# PPN-limit of Fourth Order Gravity inspired by Scalar-Tensor Gravity

###### Abstract

Based on the dynamical equivalence between higher order gravity and scalar-tensor gravity the PPN-limit of fourth order gravity is discussed. We exploit this analogy developing a fourth order gravity version of the Eddington PPN-parameters. As a result, Solar System experiments can be reconciled with higher order gravity, if physical constraints descending from experiments are fulfilled.

###### pacs:

04.50.+h, 98.80.-k, 04.25.Nx, 04.80.Cc^{†}

^{†}thanks: Corresponding author : A. Troisi,

## I Introduction

The recent debate about the origin of the cosmic acceleration, induced by the results of several astrophysical
observations sneia ; cmbr ; lss , led to investigate several theoretical approaches capable of providing viable
physical mechanisms to the dark energy problem. In this wide discussion, no scheme seems, up to now, to furnish a
final answer to this puzzling conundrum. Nevertheless among the different models, ranging from quintessential
scenarios steinhardt , which generalize the cosmological constant approach starobinsky-shani , to
higher dimensional scenarios braneworlds ; dgp or the resort to cosmological fluids with exotic equation of
state chaplygin ; vdw and unified approaches considering even dark matter hobbit ; unified , an
interesting scheme which seems to deserve a major attention is represented by higher order theories of gravity.
This approach obtained by the generalization of the Einstein gravity, has led to interesting results both in the
metric formulation curv-quint ; noi-ijmpd ; noi-review ; carroll ; odintsov-m and in the Palatini one
palatini ; francaviglia .

Recently some authors have analyzed the PPN-limit of such theories both in the metric and in the Palatini approach
olmo ; allemandi-ruggiero with contrasting results. Thus, it seems interesting to deepen the discussion about
the Post Parametrized Newtonian (PPN) behaviour of this theory. The purpose is to verify if the cosmological
reliability of such a scheme can be drawn even on the Solar System scales and to understand if the hypothesis of a
unique fluid working as a two “faces” component (matter and geometry) can be a workable one.

In this paper we exploit the strict analogy between the higher order gravity and the scalar-tensor theories to
develop a PPN-formalism for a general fourth order gravity model in the metric framework, working, in general, for
extended theories of gravity. There are strong analogies between these two approaches. The similarity between the
non-minimally coupled scalar models and the higher order gravity ones is known since 1983 teyssandier83 ,
when it was demonstrated the similarity between a scalar-tensor Lagrangian of Brans-Dicke type and fourth order
gravity. Actually, such an interpretation goes well beyond conformal transformations, since it is a formal analogy
without any physical change in the dynamical
variables of the system.

In this paper, we further discuss the analogy between fourth order gravity and scalar-tensor gravity considering
the PPN-parametrization descending from such a similarity. As main result, we show, despite some recent studies
olmo , that Solar System experiments do not exclude the possibility that higher order gravity theories can
represent a viable approach even at scales shorter than the cosmological ones. In other words, standard General
Relativity should be revised both at cosmological and Solar System distances in order to solve several mismatches
between the theoretical predictions and the observational results.

## Ii Fourth Order Gravity vs. Scalar Tensor Gravity

Let us recall how the analogies between the two schemes arise. As it is well known, scalar-tensor gravity is obtained if a scalar-field-matter Lagrangian is non - minimally coupled with the Hilbert-Einstein Lagrangian. The general action for a such theory is capoz-scal-tens :

(1) |

where is the coupling function, the self-interaction potential, a scalar field, the ordinary matter Lagrangian and the dimensional coupling. This relation naturally provides Brans-Dicke gravity brans-dicke if it is rearranged through the substitutions :

(2) |

which depends on the metric and the matter fields. Again defines the dimensional coupling.
The energy-momentum tensor of matter is given by the relation

From the action (2), we obtain the fourth order field equations : .

(3) |

which can be recast in a more expressive form as:

(4) |

where is the Einstein tensor and ; the two terms and
imply fourth order derivatives of the metric . On the other side, if is a linear
function of the scalar curvature, , the field equations become the ordinary second-order
ones.

Considering the trace of Eq.(4),

(5) |

Such an equation can be interpreted as the equation of motion of a self-interacting scalar field, where the self-interaction potential role is played by the quantity . This analogy can be developed each time one considers an analytic function of which can be algebraically inverted so that reads as , in other words it has to be . In fact, defining

(7) |

we can write Eqs.(4) and (5) as

(8) |

(9) |

which can also be obtained from a Brans-Dicke action of the form

(10) |

This expression is related to the so called O’Hanlon Lagrangian, which belongs to a class of Lagrangians
introduced in order to
achieve a covariant model for a massive dilaton theory o'hanlon .

It is evident that the Lagrangian (10) is very similar to a Brans-Dicke theory, but is lacking of the
kinetic term. The formal analogy between the Brans-Dicke scheme and fourth order gravity schemes is obtained in
the particular case .

If we consider the matter term vanishing, Eq.(5) becomes an ordinary Klein-Gordon equation,
where plays the role of an effective scalar field whose mass is determined by the self-interaction
potential.

## Iii PPN-Formalism in Scalar Tensor Gravity

Along this paper, we base our discussion on the analogy between scalar-tensor theories of gravity and the higher
order ones to analyze the problem of the PPN-limit for the fourth order gravity model. Recently the cosmological
relevance of higher order gravity has been widely demonstrated. On the other side, the low energy limit of such
theories is still not satisfactory investigated, although some results on the galactic scales have been already
achieved newtlim . A fundamental test to understand the relevance of such a scheme is to check if there is
even an accord with Solar System experiments. As outlined in the introduction, some controversial results have
been recently proposed olmo ; allemandi-ruggiero . To better develop this analysis, we can refer again to the
scalar-tensor - higher order gravity analogy, exploiting the PPN results obtained in the scalar-tensor scheme esposito-farese .

A satisfactory description of PPN limit for this kind of theories has been developed in
esposito-farese ; damour1 ; damour2 . In these works, the problem has been treated providing interesting results
even in the case of strong gravitational sources like pulsars and neutron stars where the deviations from General
Relativity are obtained in
a non-perturbative regime damour2 .
A clear summary of this formalism can be found in the papers esposito-farese and schimd05 .

The action to describe a scalar-tensor theory can be assumed, in natural units, of the form (1).
The matter Lagrangian density is again considered depending only on the metric and the matter fields.
This action can be easily redefined in term of a minimally coupled scalar field model via a conformal
transformation of the form . In fact, assuming the transformation rules:

(11) |

and

(12) |

(13) |

one gets the action

(14) |

The first consequence of such a transformation is that now the non-minimal coupling is transferred on the ordinary matter sector. In fact, the Lagrangian is dependent not only on the conformally transformed metric and the matter field but it is even characterized by the coupling function . In the same way, the field equations can be recast in the Einstein frame. The energy-momentum tensor is defined as

(15) |

establishes a measure of the coupling arising in the Einstein frame between the scalar sector and the matter one as an effect of the conformal transformation (General Relativity is recovered when this quantity vanishes). It is possible even to define a control of the variation of the coupling function through the definition of the parameter as . Regarding the effective gravitational constant, it can be expressed in term of the function

(16) |

(17) |

The above definitions imply that the PPN-parameters become dependent on the non-minimal coupling function
and its derivatives. They can be directly constrained by the observational data. Actually, Solar System
experiments give accurate indications on the ranges of ^{1}^{1}1We indicate
with the subscript the Solar System measured estimates.. Results are summarized in Tab.1.

Mercury Perih. Shift | |
---|---|

Lunar Laser Rang. | |

Very Long Bas. Int. | |

Cassini spacecraft |

The experimental results can be substantially resumed into the two limits schimd05 :

(18) |

which can be converted into constraints on and . In particular, the Cassini spacecraft value
induces the bound has to be very small, which means a very
low interaction between matter and the scalar field; conversely the second derivative can take large
values so that the matter
sector may be strongly coupled with scalar degrees of freedom esposito-farese .

Together with the Solar System experiments, even binary-pulsar tests can be physically significant to characterize
the PPN-parameters. From this analysis damour1 ; damour2 ; esposito-farese descends that the
second derivative can be a large number, i.e. even for a vanishingly small .

This constraint allows to achieve a further limit on the two PPN-parameters and ,
which can be outlined by means of the ratio : . At first sight, one
can deduce that the first derivative of the coupling function

(19) |

The singular nature of this ratio puts in evidence that it was not possible to get such a limit in the
case of weak - field experiments (see for details esposito-farese ).

For sake of completeness, we cite here even the shift that the scalar-tensor gravity induces on the theoretical
predictions for the local value of the gravitational constant as coming from Cavendish-like experiments. This
quantity represents the gravitational coupling measured when the Newton force arises between two masses :

(20) |

In the case of scalar tensor gravity, the Cavendish coupling reads :

(21) |

From the limit on coming from Cassini spacecraft, the difference between and is not more than the .

## Iv PPN limit of fourth order gravity inspired by the scalar-tensor analogy

In previous section, we discussed the PPN limit in the case of a scalar-tensor gravity. These results can be
extended to the case of fourth order exploiting the analogy with scalar-tensor case developed in Sec. 2.

We have seen that fourth order gravity is equivalent to the introduction of a scalar extra degree of freedom into
the dynamics. In particular, from this transformation, it derives a Brans-Dicke type Lagrangian with a vanishing
Brans-Dicke parameter . Performing the change of variables implied by a conformal
transformation, the Brans-Dicke Lagrangian can be furtherly transformed into a Lagrangian where the non-minimal
coupling is moved onto the matter side as in (14). The net effect is that, as in the case of a
“true” scalar-tensor theory, it is possible to develop an Einstein frame formalism which allows a PPN-limit
analysis. The basic physical difference between the two descriptions is that the quantities entering the
PPN-parameters and , or the derivatives of the non-minimal coupling function
, are now and its derivatives with respect to the Ricci
scalar since the non minimal coupling function in the Jordan frame is .

Alternatively, to obtain a more versatile equivalence between the two approaches it is possible to write down
fourth order gravity by an analytic function of the Ricci scalar considering the identification induced by the
field equations, i.e. . In fact, if one takes into account the scalar-tensor Lagrangian :

(22) |

the variation with respect and the metric provide the above identification and a system of field equations which are completely equivalent to the ordinary ones descending from fourth order gravity. The expression (22) can be recast in the form of the O’Hanlon Lagrangian (10) by means of the substitutions :

(23) |

where, in such a case, the prime means the derivative with respect to . It is evident that the new scalar-tensor description implies a non-minimal coupling function through the term

(24) |

and the identification implies that the higher order derivatives can be straightforwardly
generalized.

At this point, it is immediate to extend the results of the PPN-formalism developed for scalar-tensor gravity to
the case of a fourth order theory. In fact, it is possible to recast the PPN parameters (16)-(17)
in term of
the curvature invariants quantities.

This means that the non-minimal coupling function role, in the fourth order scenario, is played by the
quantity. As a consequence the PPN-parameters (16) and (17) become :

(25) |

(26) |

These quantities have, now, to fulfill the requirements drawn from the experimental tests resumed in Table
1. The immediate consequence of such definitions is that derivatives of fourth order gravity theories have
to satisfy constraints in relation to the actual measured values of the Ricci scalar . As a matter of fact,
one can check these quantities by the Solar System experimental prescriptions and deduce the compatibility between
fourth order gravity and General Relativity.

Since the definitions (25) and (26) do not allow to obtain, in general, upper limits on
from the constraints of Table 1, one can arbitrarily chose classes of fourth order Lagrangians, in order to check
if the approach is working. We shall adopt classes of Lagrangians which are interesting from a cosmological point
of view since give viable results to solve the dark energy problem curv-quint ; noi-review ; noi-ijmpd .

In principle, one can try to obtain some hints on the form of (or correspondently of the ) by
imposing constraints provided from the Lunar Laser Ranging (LLR) experiments and the Cassini spacecraft
measurements which give direct stringent estimates of PPN-parameters. After, one can try to solve these relations
and then to verify what is the response to the pulsar upper limit with respect to the ratio

After this remark, one can consider different fourth order Lagrangians with respect to the two Solar System
constraints coming from the perihelion shift of Mercury and the Very Long Baseline Interferometry.

The results are summarized in Table 2. We have listed the fourth order Lagrangians considered in the
first column and the limit on the model parameters induced by the Solar System constraints is in the second
column. . This procedure has shown that generally if the two
Solar System relations are verified, the pulsar constraint is well fitted by a modified gravity model. However
this result is strictly influenced by the error range of the Cassini and LLR tests.

Lagrangian | Parameters constraints |
---|---|

As it is possible to see, the PPN-limits induced by the Solar System tests can be fulfilled by different kinds of
fourth order Lagrangians provided that their parameters remain well defined with respect to the
background value of the curvature.

These results corroborate evidences for a defined PPN-limit which does not exclude higher order gravity. They are
in contrast with other recent investigations olmo ; ppn-bad , where it has been pointed out
that this kind of theories are not excluded by experimental results in the weak field limit and with respect to
the PPN prescriptions.

Similar results also hold for Lagrangians as and

## V Conclusions

Since the issue of higher order gravity is recently become a very debated matter, we have discussed its low energy
limit considering the PPN-formalism in the metric framework. The study is based on the analogy between the
scalar-tensor gravity and fourth order gravity. Such an investigation is particularly interesting even in relation
to the debate about the real meaning of the curvature fluid which could be a natural explanation for dark energy
curv-quint ; noi-review ; noi-ijmpd ; carroll ; odintsov-m ; palatini ; francaviglia ; olmo ; allemandi-ruggiero . The
PPN-limit indicates that several fourth order Lagrangians could be viable on the Solar System scales. It has to
remarked that the Solar System experiments pose rather tight constraints on the values of coupling constants, e.g.
(see Table II). Such a result does not agree with the very recent papers olmo which suggest negative
conclusions in this sense, based on questionable
theoretical assumptions and extrapolations.

It is evident that such discussion does not represent a final
answer on this puzzling issue. Nevertheless it is reasonable to
affirm that extended gravity theories cannot be ruled out, definitively, by Solar
System experiments. Of course, further accurate investigations are needed to
achieve some other significant indications in this sense, both
from theoretical and experimental points of view. For example
the study of higher order gravity PPN-limit directly in the
Jordan frame could represent an interesting task for
forthcoming investigations.

An important concluding remark is due at this point. A scalar-tensor theory can be recast in the Einstein frame,
via a conformal transformation, implying an equivalent framework. Actually, dealing with higher order
gravity, there is no more such a conformal transformation able to “equivalently” transform the whole system from
the Jordan frame to the Einstein one. Effectively, it is possible to conformally transform a higher order (and, in
particular, a fourth order) theory into an Einstein-like with the addiction of some scalar fields as a direct
consequence of the equivalence between the higher order framework and the scalar-tensor one at level of the
classical field equations. This equivalence addressed, as dynamical equivalence wands:cqg94 , does not
holds anymore when one considers configurations which do not follow the classical trajectories, for example in the
case of quantum effects. A fundamental result which follows from this considerations is that dealing with the
early-time inflationary scenario one can safely perform calculations for the primordial perturbations in the
Einstein conformal frame of a scalar-tensor model while it is not possible to develop such calculations in the
case of an higher order gravity scenario since the scalar degrees of freedom are no more independent of the
gravitational field source. This issue holds, if the effective field is induced from geometrical degrees of
freedom. Since the PPN-limit is achieved in the semi-classical limit, when the conformal factor turns out to be
well defined, deductions about the PPN-limit for fourth order gravity models, developed exploiting the analogy
with the scalar-tensor scheme, are safe from problems.

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